Fourier Analysis On Number Field And Computation Of Modular Forms

In this paper, we generalize the classical Riemannζ-function to L-function which is the sum over integral ideals of a certain type of ideal character. Also we can generalize it to Hecke L-function, which is based on the adele ring and the idele group associated with a number field. It's well known that E. Hecke has established the analytic contin-uation and the functional equation of L-function for any idele class character x by an enormously complicated application of generalized theta functions and computation. In this paper, we introduce Tate's method to solve these kinds of problems. Tate used a simple Poisson summation formula for general functions of adele, summed over the discrete subgroup of field elements. With this Poisson summation formula, which is of great importance in itself, in as much as it is the number theoretic analogue of the Riemamm-Roch theorem, an analytic continuation of the generalized L-functions can be given, and an elegant functional equation can be established for them. The content of this paper contains 8 chapters as follows:In Chapter 1, we have proved that Riemannζ-function has analytic continuation to complex plane and explicit functional equation;In Chapter 2 and Chapter 4, we introduced the definition of local field, Adele ring, Idele group and some basic results of them;In Chapter 3, after showing the analysis structure of any local field, such as Haar measure, characters and the local L-factors associated with the character, we also de-fined the Schwartz-Bruhat function f on a local field and the localζ-function of f. Moreover, we proved there is analytic continuation of such localζ-function and good relation between localζ-functions and the local L-factors before, before;In Chapter 5, we first generalize the Schwartz-Bruhat function to adelic Schwartz-Bruhat function on the Adele ring of any number field K. Then we had proved Poisson summation formula and Riemann-Roch theorem over any number field K associated with the adelic Schwartz-Bruhat function.In Chapter 6, we defined the globalζ-function over the Idele group of K of the adelic Schwartz-Bruhat function associated with the idele class character X. Then we prove that the globalζ-function has analytic continuation and good functional equation by Riemann-Roch theorem.In Chapter 7, the key theorem of this paper has been proved. That is, Hecke L-function of any idele class character x associated with any number field K, which is a gener-alization of Riemannζ-function, can analytic continue to complex plane and exists excellent functional equation. This is mainly because that there is an explicit adelic Schwartz-Bruhat function which relates Hecke L-function with the globalζ-function by the result of Chapter 3. Finally, we acquire the analytic continuation and functional equation of Hecke L-function from the same properties of the globalζ-function. In the last chapter, there is nice relation between modular forms and idele class charac-ter X.We introduced the definition of eta-quotient and computed the fourier expansion of two eta-quotients, which is very explicit to modular forms.